Chaos and scrambling in simple quantum mechanical systems via OTOC - - PowerPoint PPT Presentation
#22 Chaos and scrambling in simple quantum mechanical systems via OTOC Ryota Watanabe (Osaka University) T. Akutagawa, K. Hashimoto, T. Sasaki, R. Watanabe [2004.04381] K. Hashimoto, K. B. Huh, K. Y. Kim, R. Watanabe [2007.04746] 1/14
#22 1/14
2/14
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Level Statistics OTOC
spacing
[Maldacena, Shenker, Stanford `16]
[Xu, Scaffidi, Cao `19]
[Larkin, Ovchinnikov `69] [Kitaev `14]
4/14
A coupled harmonic oscillator An inverted harmonic oscillator
[Matinyan, Savvidy, T. A. Savvidy `81]
[Matinyan, Savvidy, T. A. Savvidy `81] [Haller, Köppel, Cederbaum `84]
VOLUME 52, NUMBER 19
PHYSICAL REVIEW LETTERS
7 MAY 1984
35
&E
&85
292
Levels
q= 0.271
50—
N I I I I85
& E &135
(U569
Levels
(D50—
K
E
z
q= 0.508
175
& E &225
608 Levels
= 0.812
50—
time on an IBM 370/168 utilizing thereby
almost
all
In order to perform a statistical
analysis of a given
spectral sequence
appropriately
it is exceedingly im- portant
to have a constant
average
spacing between
neighboring energy
levels26
since global
variations
can falsify the fluctuation mea- sures considerably. Various
unfolding
procedures are known to decompose a given spectrum into sec-
ular variations and fluctuations.
26 27 Here we have
used cubic spline for smoothing
with an appropriate
number
the secular
variations
level density of the spectrum of H
[Eq. (1)] in a given energy
interval. A second im- portant
point is the necessity of large data sets in or-
der to obtain reasonable fluctuation
measures.
For
each
energy interval investigated
we
consider
—
200 energy
levels
as a lower limit for a reliable
statistical test.
The distributions
between
adja-
cent levels of the unfolded
spectral
sequence
are
displayed
as histograms in Fig. 1 for specified ener-
gy intervals.
The abscissa is in units of D which
is
the average
spacing
between adjacent levels
in the
respective
energy
interval.
Here
a value
coupling
parameter
An obvious
trend
from a Poisson-type
distribution
in
the
low-lying
energy interval
[Fig. (la)]
to
Wigner-type distributions
in the high-energy
regime
[Figs. 1(b) and 1(c)] can be seen. We are able to
approximate the histograms conveniently
by a con-
tinuous distribution
function
Pq(S)
specified
by
the parameter
q:
P,(S)=
S~exp( —
PSt+q);
n = (1+q)P,
P = [D-tr [(2+q)/(1+ q)]]'+q. (2)
For
q =0 P~(S) recovers
the Poisson
distribution
Pp(S) = (1/D )exp( — S/D ),
and
for
q = 1
the
Wigner distribution
Pt(S) = (n S/2D2)exp( —
mS2/
4D2). It was introduced
first by Brody28 in order to fit NNS histograms
from nuclear spectra.
The respective
values
in Fig. 1 for
the different
energy intervals have been calculated
by a least-squares
fit.
Having
the transition from regularity
to
irregularity
for a specific Hamiltonian
we now vary
the
coupling
parameter
k. Figure
2 shows in its
upper part the variation
q parameters
with
energy for different values of k [Eq. (1)]. For their
computation
we used energy
intervals
50
whose
centers are the abscissae
data
points.
A continuation
to the
right would
require
more converged
eigenvalues. However,
it is evident
that
q will tend to unity
as
the energy and/or
the value of the coupling
param-
eter k increase. The curves have been truncated
at
the left-hand side, since the
histograms
for low-
lying energy
intervals
exhibit the anomalies charac- teristic
systems
harmonic
as analyzed thoroughly
by Berry and Tabor. 2o
In order
to elucidate the functional dependence
q on the energy E and the
coupling
parameter
k we employ
the scaling proper-
ty of the classical Hamilton
function
histograms
for ener-
gy levels of the Pullen-Edmonds
Hamiltonian
[Eq. (1)]
with k = 0.005. The results for three different
energy in- tervals
are shown. In each case a best-fitting Brody dis-
tribution
[Eq. (2)] specified
by the parameter
q is also
shown.
= (1/k) H(1;ptJk, qtJk, p2Jk,
q2jk ),
(3)
where H(k; pt, qt, p2, q2) is the classical Hamilton
function corresponding to the quantum
mechanical Hamiltonian
Hof Eq. (1). It means that the classi-
cal dynamics
depends
solely
product
kE. Therefore,
we have plotted in Fig. 2(b) the Brody
1666
Energy 5/14
6/14
1 2 3 4 5 5 10 15 20 25 30
S/D
<latexit sha1_base64="6JaxiO2fIrtxgdBxQkYq8nZ3bM=">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</latexit>Exponential growth Oscillatory
Eigenstates totally symmetric under 𝐷!" symmetry [Akutagawa, Hashimoto, Sasaki, Watanabe 2004.04381]
𝐸: the average spacing Exponential growth seen in OTOC Chaos not seen in level statistics 7/14
Classical Quantum OTOC Level statistics Regular Mixed Chaotic Oscillatory Exponential growth Non-Wigner Wigner
E or T
1 5 4 3 2
[Akutagawa, Hashimoto, Sasaki, Watanabe 2004.04381]
8/14
9/14
[Hashimoto, Murata, Yoshii `17]
10/14
1 2 3 4
2 4 t ln[cn(t)] n=1 n=5 n=9 n=11 n=13 n=20
Exponential growth Oscillatory
5 10 x 10 20 30 40 V(x)
The exponential growth originates from the local instability, not chaos.
[Hashimoto, Huh, Kim, Watanabe 2007.04746]
11/14 The potential and the levels Red: growth is seen Microcanonical OTOC
1 2 3 4
1 2 3 4 t ln[CT(t)] T=1 T=5 T=9 T=30
2.1 2.0 1.9 1.8 1.7
!"#"$
2.2 1.6 10 20 30 40 50 60 70 80
%
Exponential growth Oscillatory
is non-vanishing in
[Hashimoto, Huh, Kim, Watanabe 2007.04746]
12/14 Thermal OTOC Temperature dependence of the Lyapunov exponent
Semiclassically, is bounded by the classical Lyapunov from below [Xu, Scaffidi, Cao `19].
13/14
14/14
1 0.5
0.5 1
1 2
1 0.5 5
2 4
Regular (E=1) Mixed (E=2) Chaos (E=20) Poincaré sections
[Matinyan, Savvidy, T. A. Savvidy `81] [Akutagawa, Hashimoto, Sasaki, Watanabe 2004.04381]
[Hashimoto, Murata, Yoshii `17]
. . .
.
[Akutagawa, Hashimoto, Sasaki, Watanabe 2004.04381]
.
Poisson distribution Wigner distribution
[Oganesyan, Huse `07] [Atas, Bogomolny, Giraud, Roux `13]
1 2 3 4 5 5 10 15 20 25 30
S/D
<latexit sha1_base64="6JaxiO2fIrtxgdBxQkYq8nZ3bM=">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</latexit>1 2 3 4 5 5 10 15 20 25 30
S/D
<latexit sha1_base64="6JaxiO2fIrtxgdBxQkYq8nZ3bM=">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</latexit>35 eigenenergies 100 eigenenergies
[Akutagawa, Hashimoto, Sasaki, Watanabe 2004.04381]